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97 . Suppose that an accounting firm does randomly surveys 100 study to determine the time needed people_ The sample mean is 23.6 hours. There complete one person &...

Question

97 . Suppose that an accounting firm does randomly surveys 100 study to determine the time needed people_ The sample mean is 23.6 hours. There complete one person '$ tax foris: population distribution assumed t0 be normal, known standard deviation of 7.0 hours. TheIn words, clefine the random variables X and Which distribution should YOu use for this problem? Explain your choice Construct 90% confidence interval for the population mean time comiplete the tax forns: State the confidence inte

97 . Suppose that an accounting firm does randomly surveys 100 study to determine the time needed people_ The sample mean is 23.6 hours. There complete one person '$ tax foris: population distribution assumed t0 be normal, known standard deviation of 7.0 hours. The In words, clefine the random variables X and Which distribution should YOu use for this problem? Explain your choice Construct 90% confidence interval for the population mean time comiplete the tax forns: State the confidence interval Sketch the graph Calculate the error bound,



Answers

Suppose that an accounting firm does a study to determine the time needed to complete one person's tax forms. It randomly surveys 100 people. The sample mean is 23.6 hours. There is a known standard deviation of 7.0 hours. The population distribution is assumed to be normal.
a.
i. $\overline{x}=$ _____
ii. $\sigma=$ _____
iii. $n=$ _____
b. In words, define the random variables $X$ and $\overline{X}$ .
c. Which distribution should you use for this problem? Explain your choice.
d. Construct a 90$\%$ confidence interval for the population mean time to complete the tax forms.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
e. If the firm wished to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make?
f. If the firm did another survey, kept the error bound the same, and only surveyed 49 people, what would happen to the level of confidence? Why?
g. Suppose that the firm decided that it needed to be at least 96% confident of the population mean length of time to within one hour. How would the number of people the firm surveys change? Why?

The following is a solution of number 13 and this looks at audited estate tax return. And we have a random sample of 100 estate tax returns that were audited by the I. R. S. And the sample mean for additional taxes owed on those returns is $3137. And it says the population standard deviation of all A state tax returns, the amount that they always 26-94. So we're actually given this sigma here and we're asked to find the 90% confidence interval for the mean amount owed on a state tax returns after their audited. So we're gonna use the Z. Interval here. And the reason why we're able to use a Z. Interval is because we actually know what the population standard deviation is, we know what sigma is. Therefore we're allowed to use the Z. Interval. And I'm going to use you can use the formula if you want or any sort of technology. I'm gonna use the T. I. 84 because it works out pretty nicely if you go to stat and then air over two tests, it's this seventh option down here Z in interval and just make sure summary stats is highlighted there And then we can start filling up what we know. So the sigma was 26-94, that's the population standard deviation. And the sample mean was 31 37 And the sample size was 100. And then we want to be 90% confidence of nine is our sea level and then here's your X bar on your end. But this top band is really what we want. So 26 93.9 and 35 80.1 is our confidence intervals. So let's go and write that down. 26 93.9 I'll go and say 90 since we're doing it in dollars and 35 80 0.10. Okay, so that's the confidence interval here and then this one, we're actually supposed to interpret it. So I'm gonna interpret it and I'm actually gonna write it down or type it in just so it'll be neater and a little bit more quick, little quicker. But the way that we interpret these, they all kind of follow the same structure. We're going to say we are 90% confident that the mean tax amount, Oh dude, for all estate tax returns that have been audited is between 26 93 9. Let's put a dollar amount in front of it just in case. So $2,693.90 and $3,580 and 10 cents. So we are 90% confident that the mean tax amount owed after an audit on a state tax return is between 26 94 basically and 35 $80

Using the following data identified in the bottom line here, we want to find the sample mean X. Bar the sample standard deviation S. And accordingly constructed 99% confidence interval for the population. Mean you assuming this population is normally distributed To start off with, let's find expire and s using the appropriate definitions expire is simply that some of the data divided by n or 7.487 S is simply the square root of the sum of deviations about the main square, divided by n minus one. In this case. 1.638 next. Since we have a sample standard deviation, not the population, we're going to use the student's T distribution to calculate this confidence interval. The T score we need to identify is that where the students distribution area following between our negative and positive critical value TC is 0.99 for a degree of freedom and minus 24 15. Using a tea table in google or textbook, we obtained T C equals 2.947 Now we need to construct the margin of error E, which is given by TC times sample standard deviation asked, divided by route and plugging in our T C, s and N gives E equals 1.207 And now we can construct a confidence interval using the following formula that mu falls between expo minus E and explore plus key, meaning that our confidence interval is new, following between 6.280 and 8.694 with 99% confidence.

Crap. Using the data below. We want to find sample, the ex far sample standard deviation S. And according to construct a 99% confidence interval for the data's population, assuming the population is normally distributed first, let's find X. Bar and S. Using their definitions X bar is the some of the data divided by M. Or in this case 12.194 and S. Is the square root of the sum of deviations about the main square, all divided by n minus one. Or in this case 1.75 Now, since we only have a sample sender view, we should not a population standard deviation. We're going to use the students to distribution to identify this confidence interval. So we need to identify a T. Score for degree of freedom 17 where the probability that she falls between the negative and positive, critical values 170.99 Using a tea table, we obtain T. C equals 2.898 Next to construct the margin of error. E. Given by the following formula plugging in our T C. S and M. We obtain E equals 1.199 And now we construct a confidence interval about mu. Given by this formula, which allows us to derive the final solution. 10.995 is less than video, is less than 13.393 with 99% confidence.

The following is a solution in # nine, and this says that we have a random sample of 40. Now, it doesn't say the population distribution, but it doesn't matter because that sample size is big enough to assume normality and assume the central limit theorem. So, had this been a Sample of like size 20 or 25, Then we would actually have to say, but it comes from a normal population, but since that sample sizes 40, it doesn't matter how skewed or or whatever the population is, 40 is big enough. And the means of that sample is 120.5, and the standard deviation of that sample is 12.9, and were asked to find the 99% confidence interval for the population mean, μ So first off, let's decide what method to use. We're gonna use the tea interval here. So the T interval. The reason why we use the tea interval is because we're estimating the population mean, So it's either going to be the Z. Or the tea interval. And the reason why we can't use the Z. Interval is because we don't know what sigma is, we don't know that population standard deviation. We only know the sample standard deviation S. So whenever you don't know that sigma, you don't know that population standard deviation, you have to use the T and a rule. So you can use the tea interval formula or you can just use technology, I'm going to use the calculator. The T. IATA four calculate because it works out pretty nice. So if you go to stat and then tests it's his eighth option down here at the T interval, so click eight and then summary stats should be highlighted there and then we can start punching in our numbers, so X. Bar for number nine was 1 20.5, and then the sample standard deviation was 12 9, And then the sample size was 40. And again we were asked to find the 99% confidence and it was a .99 would be our confidence level and then we calculate, and then this top band here, that's going to be Our confidence interval. So 1 14.98-1 26.02. So let's go and write that down. So 1 14 9, 8 To 1 26.02. Okay, so it doesn't say to interpret this, but if you were to interpret even though there's no context to this problem, we would just say we are 99% confident that the true population mean is between 1 14.98 and 1 26.2


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